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Course Queries Syllabus Queries 2 years ago
Posted on 16 Aug 2022, this text provides information on Syllabus Queries related to Course Queries. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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There is a theorem in my book which says the following:
Let KK be a field and let f(X)∈K[X]f(X)∈K[X] be irreducible over KK. Then there exists a field extension L/KL/K, such that ∃u∈L∃u∈L: f(u)=0f(u)=0.
Proof: Notice that K⊆K[X]/(f(x))K⊆K[X]/(f(x)). Clearly, K[X]/(f(x))K[X]/(f(x)) is a field since (f(x))(f(x)) is maximum ideal. If we take u=X¯¯¯¯u=X¯, then f(u)=f(X¯¯
X¯¯¯¯X¯ means the class of XX mod f(X)f(X) in the quotient ring K[X]/(f(x))K[X]/(f(x)).
Now, given g∈K[X]g∈K[X], consider the associated polynomial function in K[X]/(f(x))K[X]/(f(x)). Then, by the definition of the ring operations in K[X]/(f(x))K[X]/(f(x)), we have g(u)=g(X¯¯¯¯)=g(X)¯¯¯¯¯¯¯¯¯font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: no
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